Penney's game for permutations
arxiv(2024)
摘要
We consider the permutation analogue of Penney's game for words. Two players,
in order, each choose a permutation of length k≥3; then a sequence of
independent random values from a continuous distribution is generated, until
the relative order of the last k numbers coincides with one of the chosen
permutations, making that player the winner.
We compute the winning probabilities for all pairs of permutations of length
3 and some pairs of length 4, showing that, as in the original version for
words, the game is non-transitive. Our proofs introduce new bijections for
consecutive patterns in permutations. We also give some formulas to compute the
winning probabilities more generally, and conjecture a winning strategy for the
second player when k is arbitrary.
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